Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra

We prove that a finite dimensional algebra $A$ with representation-finite subcategory consisting of modules that are semi-Gorenstein-projective and $n$-th syzygy modules is left weakly Gorenstein. This generalises a theorem of Ringel and Zhang who proved the result in the case $n=1$. As an application we show that monomial algebras and endomorphism rings of modules over representation-finite algebras are weakly Gorenstein. We then give a new connection between the theory of dominant dimension and Gorenstein homological algebra for gendo-symmetric algebras. As an application, we will see that the existence of a non-projective Gorenstein-projective-injective module in a gendo-symmetric algebra already implies that this algebra is not CM-finite. We apply out methods to give a first systematic construction of non-weakly Gorenstein algebras using the theory of gendo-symmetric algebras. In particular, we can construct non-weakly Gorenstein algebras with an arbitrary number of simple modules from certain quantum exterior algebras such as the Liu-Schulz algebra.